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Quantum Computation

A quantum computer (also known as a quantum supercomputer) is a computation device that makes direct use ofquantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from digital computers based on transistors. Whereas digital computers require data to be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses qubits(quantum bits), which can be in superpositions of states. A theoretical model is the quantum Turing machine, also known as the universal quantum computer. Quantum computers share theoretical similarities with non-deterministic and probabilistic computers; one example is the ability to be in more than one state simultaneously. The field of quantum computing was first introduced by Yuri Manin in 1980[2] and Richard Feynman in 1982. A quantum computer with spins as quantum bits was also formulated for use as a quantum space–time in 1969.

As of 2014 quantum computing is still in its infancy but experiments have been carried out in which quantum computational operations were executed on a very small number of qubits. Both practical and theoretical research continues, and many national governments and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as cryptanalysis.

Large-scale quantum computers will be able to solve certain problems much more quickly than any classical computer using the best currently known algorithms, like integer factorization using Shor’s algorithm or the simulation of quantum many-body systems. There exist quantum algorithms, such as Simon’s algorithm, which run faster than any possible probabilistic classical algorithm. Given sufficient computational resources, however, a classical computer could be made to simulate any quantum algorithm; quantum computation does not violate the Church–Turing thesis.

Entanglement is a term used in quantum theory to describe the way that particles of energy/matter can become correlated to predictably interact with each other regardless of how far apart they are. Particles, such as photons, electrons, or qubits that have interacted with each other retain a type of connection and can be entangled with each other in pairs, in the process known as correlation. Knowing the spin state of one entangled particle – whether the direction of the spin is up or down – allows one to know that the spin of its mate is in the opposite direction. Even more amazing is the knowledge that, due to the phenomenon of superposition, the measured particle has no single spin direction before being measured, but is simultaneously in both a spin-up and spin-down state. The spin state of the particle being measured is decided at the time of measurement and communicated to the correlated particle, which simultaneously assumes the opposite spin direction to that of the measured particle. Quantum entanglement allows qubits that are separated by incredible distances to interact with each other immediately, in a communication that is not limited to the speed of light. No matter how great the distance between the correlated particles, they will remain entangled as long as they are isolated.

Entanglement is a real phenomenon (Einstein called it “spooky action at a distance”), which has been demonstrated repeatedly through experimentation. The mechanism behind it cannot, as yet, be fully explained by any theory. One proposed theory suggests that all particles on earth were once compacted tightly together and, as a consequence, maintain a connectedness. Much current research is focusing on how to harness the potential of entanglement in developing systems for quantum cryptography and quantum computing. In 1997, Nicholas Gisin and colleagues at the University of Geneva used entangled photons to enable simple – but instantaneous – communication over a distance of seven miles.

In quantum computing, a qubit (/ˈkjuːbɪt/) or quantum bit is a unit of quantum information—the quantum analogue of the classical bit. A qubit is a two-state quantum-mechanical system, such as the polarization of a single photon: here the two states are vertical polarization and horizontal polarization. In a classical system, a bit would have to be in one state or the other, but quantum mechanics allows the qubit to be in a superposition of both states at the same time, a property which is fundamental to quantum computing.

The two states in which a qubit may be measured are known as basis states (or basis vectors). As is the tradition with any sort of quantum states, they are represented by Dirac—or “bra–ket”—notation. This means that the two computational basis states are conventionally written as and (pronounced “ket 0” and “ket 1”).

A pure qubit state is a linear superposition of the basis states. This means that the qubit can be represented as a linear combination of and : where α and β are probability amplitudes and can in general both be complex numbers. When we measure this qubit in the standard basis, the probability of outcome is and the probability of outcome is. Because the absolute squares of the amplitudes equate to probabilities, it follows that α and β must be constrained by the equation simply because this ensures you must measure either one state or the other.

In quantum computing and specifically the quantum circuit model of computation, a quantum gate (or quantum logic gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.

Unlike many classical logic gates, quantum logic gates are reversible. However, classical computing can be performed using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions. This gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits.

Quantum logic gates are represented by unitary matrices. The most common quantum gates operate on spaces of one or two qubits, just like the common classical logic gates operate on one or two bits. This means that as matrices, quantum gates can be described by 2 × 2 or 4 × 4 unitary matrices.

Shor’s algorithm, named after mathematician Peter Shor, is a quantum algorithm (an algorithm that runs on a quantum computer) for integer factorizationformulated in 1994. Informally it solves the following problem: Given an integer N, find its prime factors. On a quantum computer, to factor an integer N, Shor’s algorithm runs in polynomial time (the time taken is polynomial in log N, which is the size of the input). Specifically it takes time O((log N)3), demonstrating that the integer factorization problem can be efficiently solved on a quantum computer and is thus in thecomplexity class BQP. This is substantially faster than the most efficient known classical factoring algorithm, the general number field sieve, which works in sub-exponential time — about O(e1.9 (log N)1/3 (log log N)2/3). The efficiency of Shor’s algorithm is due to the efficiency of the quantum Fourier transform, and modular exponentiation by repeated squarings.

If a quantum computer with a sufficient number of qubits could operate without succumbing to noise and other quantum decoherence phenomena, Shor’s algorithm could be used to break public-key cryptography schemes such as the widely used RSA scheme. RSA is based on the assumption that factoring large numbers is computationally infeasible. So far as is known, this assumption is valid for classical (non-quantum) computers; no classical algorithm is known that can factor in polynomial time. However, Shor’s algorithm shows that factoring is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer. It was also a powerful motivator for the design and construction of quantum computers and for the study of new quantum computer algorithms. It has also facilitated research on new cryptosystems that are secure from quantum computers, collectively called post-quantum cryptography.

In 2001, Shor’s algorithm was demonstrated by a group at IBM, who factored 15 into 3 × 5, using an NMR implementation of a quantum computer with 7 qubits. However, some doubts have been raised as to whether IBM’s experiment was a true demonstration of quantum computation, since no entanglement was observed. Since IBM’s implementation, several other groups have implemented Shor’s algorithm using photonic qubits, emphasizing that entanglement was observed. In 2012, the factorization of 15 was repeated. Also in 2012, the factorization of 21 was achieved, setting the record for the largest number factored with a quantum computer. In April 2012, the factorization of 143 was achieved, although this used adiabatic quantum computation rather than Shor’s algorithm.